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{point.vals}"}}],"xAxis":{"type":"linear","crosshair":true,"title":{"text":"Time t"}}},"theme":{"chart":{"backgroundColor":"transparent"},"colors":["#7cb5ec","#434348","#90ed7d","#f7a35c","#8085e9","#f15c80","#e4d354","#2b908f","#f45b5b","#91e8e1"]},"conf_opts":{"global":{"Date":null,"VMLRadialGradientURL":"http =//code.highcharts.com/list(version)/gfx/vml-radial-gradient.png","canvasToolsURL":"http =//code.highcharts.com/list(version)/modules/canvas-tools.js","getTimezoneOffset":null,"timezoneOffset":0,"useUTC":true},"lang":{"contextButtonTitle":"Chart context menu","decimalPoint":".","downloadJPEG":"Download JPEG image","downloadPDF":"Download PDF document","downloadPNG":"Download PNG image","downloadSVG":"Download SVG vector image","drillUpText":"Back to {series.name}","invalidDate":null,"loading":"Loading...","months":["January","February","March","April","May","June","July","August","September","October","November","December"],"noData":"No data to display","numericSymbols":["k","M","G","T","P","E"],"printChart":"Print chart","resetZoom":"Reset zoom","resetZoomTitle":"Reset zoom level 1:1","shortMonths":["Jan","Feb","Mar","Apr","May","Jun","Jul","Aug","Sep","Oct","Nov","Dec"],"thousandsSep":" ","weekdays":["Sunday","Monday","Tuesday","Wednesday","Thursday","Friday","Saturday"]}},"type":"chart","fonts":[],"debug":false},"evals":[],"jsHooks":[]}</script> ] .pull-right[ The CDF is formally defined as the cumulative probability that the R.V. <span class="mjx-char MJXc-TeX-math-I" style="display:inline;">T</span> is equal to or less than <span class="mjx-char MJXc-TeX-math-I" style="display:inline;">t</span> $$ F(t) = \Pr(T \le t) $$ Less formally, `\(F(t)\)` is equal to the area under the curve of the density function in the interval `\([0,t]\)`, i.e. $$ F(t) = \int_{0}^{t}f(u)du $$ The Weibull CDF is expressed as $$ F(t) = 1-\exp\left[-\left(\frac{t}{\eta}\right)^{\beta}\right] $$ ] <!--end .pull-right --> ] <!--end .panel --> .panel[.panel-name[CDF/PDF Relationship] <img src="data:image/png;base64,#images/cdf_pdf.gif" style="display: block; margin: auto;" /> ] ] <!--end .panelset --> --- ## stuff - <span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.472em; padding-bottom: 0.259em; padding-right: 0.106em;">F</span> - <span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.472em; padding-bottom: 0.259em; padding-right: 0.106em;">(t,&infin;)</span> --- ## Generating the plotted points .panelset[ .panel[.panel-name[Overview] Regardless of how the axes are drawn, the plotted points are computed in a similar manner - X-axis: observed event times (failure, suspension, or other event) - Y-axis: nonparametric estimate of the failure probability (CDF) at the observed event times The nonparametric estimate of the CDF `\(\widehat{F}\)` plays a key role in probability plotting - One cannot observe the reliability or unreliability of a item - events are the only observable source of information - Reliability or unreliability values must be estimated from the data - The following panels discuss some considerations for computing the values of the plotted points ] .panel[.panel-name[X coordinates] .pull-left[ The x-coordinate of each plotted point is determined by how the observed event is categorized - Exact failure: "exact" failure time observed `\(t_f = t\)` - Left censored: failure is discovered at first inspection - exact failure time not known `\(t_f \in (0,t_{1})\)` - Interval censored: failure occurs between inspections and is discovered at inspection `\(i=2,\cdots,n\)` - exact failure time not known `\(t_f \in (t_{i},t_{i+1}), i > 0\)` - Right censored: failure not observed at final `\(n^{th}\)` inspection - exact failure time not known `\(t_f \in (t_{n},\infty\right)\)` Data sets that include only exact failures are called complete data sets ] .pull-right[ <img src="data:image/png;base64,#plotly_test_files/figure-html/unnamed-chunk-6-1.png" style="display: block; margin: auto;" /> ] ] .panel[.panel-name[Y coordinates] As previously stated, the y-coordinates of the plotted points are nonparametric estimates of the CDF corresponding to the observed event times Many different estimators have been developed for this purpose, some of which are - Plotting position estimators - Kaplan-Meier estimator (aka the product limit estimator) - Turnbull Estimator The choice of which estimator to use is driven by the type censoring (suspensions) that present in the data ] .panel[.panel-name[Plotting Positions] Probability plotting positions express the non-exceedance probability of the CDF for the `\(i^{th}\)` ascending data value The generic plotting position formula is expressed as $$ \widehat{F(t_{i})}=\frac{i-a}{n+1-2a} $$ - where + `\(i\)` is an index of the ordered observations (smallest `\(\rightarrow\)` largest) + `\(n\)` is the number of observations + `\(a\)` is the <span class="explain">plotting position parameter</span><span class="tooltip">The value of `\(a\)` is chosen to produce approximately unbiased estimates of `\(F(t_{i})\)` for an assumed distribution</span> Various formulae <span class="explain">have been developed</span> <span class="tooltip"> <br/> Author | Year | `\(a\)` | Formula -----------|------|-------|--------------- Hazen | 1914 | 0.50 | `\(\displaystyle\frac{i-0.5}{n}\)` Weibull | 1939 | 0 | `\(\displaystyle\frac{i}{n+1}\)` Blom | 1958 | 0.375 | `\(\displaystyle\frac{i-0.375}{n+0.25}\)` Gringorten | 1963 | 0.44 | `\(\displaystyle\frac{i-0.44}{n+0.12}\)` Chegodayev | 2000 | 0.30 | `\(\displaystyle\frac{i-0.30}{n+0.4}\)` Cunnane | 1977 | 0.40 | `\(\displaystyle\frac{i-0.40}{n+0.2}\)` Median | 1943 | 0.3175| `\(\displaystyle\frac{i-0.3175}{n+0.365}\)` <br/> The formulae used most often in practice are: <br/> $$ `\begin{aligned} \text{Hazen}:\; &\widehat{F(t_{i})}=\frac{i-0.5}{n}\\\\ \text{Chegodayev}:\; &\widehat{F(t_{i})}=\frac{i-0.3}{n+0.4}\\\\ \text{Weibull}:\; &\widehat{F(t_{i})}=\frac{i}{n+1}\\\\ \end{aligned}` $$ <br/> Note: the Chegodayev plotting position is also known as "Median-Ranks" </span> to correspond with specific distributions ] ] --- ## table Author | Year | `\(a\)` | Formula -----------|------|-------|--------------- Hazen | 1914 | 0.50 | `\(\displaystyle\frac{i-0.5}{n}\)` Weibull | 1939 | 0 | `\(\displaystyle\frac{i}{n+1}\)` Blom | 1958 | 0.375 | `\(\displaystyle\frac{i-0.375}{n+0.25}\)` Gringorten | 1963 | 0.44 | `\(\displaystyle\frac{i-0.44}{n+0.12}\)` Chegodayev | 2000 | 0.30 | `\(\displaystyle\frac{i-0.30}{n+0.4}\)` Cunnane | 1977 | 0.40 | `\(\displaystyle\frac{i-0.40}{n+0.2}\)` Median | 1943 | 0.3175| `\(\displaystyle\frac{i-0.3175}{n+0.365}\)` --- ## Distribution Functions - Quantile .panelset[.panel[.panel-name[Quantile Function] .pull-left[ <div id="htmlwidget-1d58bc80fe76b0d33b6a" style="width:440px;height:400px;" class="highchart html-widget"></div> <script type="application/json" 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